Institute of Southern Punjab, Multan

Transcription

1 Institute of Southern Punjab, Multan Mr. Muhammad Nouman Farooq BSC-H (Computer Science) MS (Telecomm. and Networks) Honors: Magna Cumm Laude Honors Degree Gold Medalist! Blog Url: noumanfarooqatisp.wordpress.com

2 Advance Database Systems Relational Algebra & Calculus Lecture# 5

3 Lecture 5: Relational Algebra & Calculus Relational Algebra Relational Calculus 3

4 Relational Algebra 4

5 Relational Algebra It is a language in which we can ask questions (query) to a database. The relational algebra is a theoretical language with operations that work on one or more relations to define another relation without changing the original relation(s). There are many variations of the operations that are included in relational algebra. Codd (1972) originally proposed eight operations, but several others have been developed. The five fundamental operations in relational algebra are Selection, Projection, Cartesian product, Union, and Set difference, perform most of the data retrieval operations that we are interested in. 5

6 Continued In addition, there are also the Join, Intersection, and Division Operations, which can be expressed in terms of the five basic operations. The function of each operation is illustrated in Figure 4.1 on next slide. The Selection and Projection operations are Unary Operations, since they operate on one relation. The other operations work on pairs of relations and are therefore called Binary Operations. In the following definitions on next slides, let R and S be two relations/tables defined over the attributes: A = (a1, a2,..., an) and B = (b1, b2,..., bm), respectively. 6

7 Continued 1. Unary Operations: 1) Selection Operation 2) Projection Operation 2. Set Operations: 1) Union Operation 2) Set Difference Operation 3) Intersection Operation 4) Cartesian Product Operation 7

8 Continued 3. Join Operations: 1) Theta/Equi Join Operation 2) Natural Join Operation 3) Left Outer Join Operation 4) Semi Join Operation 4. Division Operations: 1) Division Operation 5. Aggregate & Grouping Operations: 1. Aggregate Operation 2. Grouping Operation 8

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10 1. Unary Operations: 1) Selection or Restriction Operation: - 10

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13 2) Projection Operation: - 13

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16 2. Set Operations: 1) Union Operation: - 16

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19 2) Set Difference Operation: - 19

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22 3) Intersection Operation: - 22

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27 4) Cartesian Product Operation: - 27

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32 3. Join Operations: 1) Theta/Equi Join Operation: - 32

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34 2) Natural Join Operation: - 34

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37 3) Left Outer Join Operation: - 37

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40 4) Semi Join Operation: - 40

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43 4. Division Operations: 1) Division Operation: - 43

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46 5. Aggregate & Grouping Operations: 1) Aggregate Operation: - 46

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49 2) Grouping Operation: - 49

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53 Relational Calculus 53

54 Relational Calculus In the relational calculus, there is no description of how to evaluate a query; a relational calculus query specifies what is to be retrieved rather than how to retrieve it. The relational calculus is not related to differential and integral calculus in mathematics, but takes its name from a branch of symbolic logic called predicate calculus. When applied to databases, it is found in two forms: tuple relational calculus, as originally proposed by Codd (1972), and domain relational calculus, as proposed by Lacroix and Pirotte (1977). 54

55 Continued In first-order logic or predicate calculus, a predicate is a truthvalued function with arguments. When we substitute values for the arguments, the function yields an expression, called a proposition, which can be either true or false. For example, the sentences, John White is a member of staff and John White earns more than Ann Beech are both propositions, since we can determine whether they are true or false. In the first case, we have a function, is a member of staff, with one argument (John White); in the second case, we have a function, earns more than, with two arguments (John White and Ann Beech). 55

56 Continued If a predicate contains a variable, as in x is a member of staff, there must be an associated rangefor x. When we substitute some values of this range for x, the proposition may be true; for other values, it may be false. For example, if the range is the set of all people and we replace x by John White, the proposition John White is a member of staff is true. If we replace x by the name of a person who is not a member of staff, the proposition is false. 56

57 Continued If P is a Predicate/Condition, then we can write the set of all x such that P is true for x, as: {x P(x)} We may connect predicates by the logical connectives: (AND), (OR), and ~ (NOT) to form compound predicates 57

58 Tuple Relational Calculus In the tuple relational calculus we are interested in finding tuples for which a predicate is true. The calculus is based on the use of tuple variables. A tuple variable is a variable that ranges over a named relation: that is, a variable whose only allowable values are tuples of the relation. For example, to specify the range of a tuple variable S as the Staff relation, we write: Staff(S) To express the query Find the set of all tuples S such that F(S) is true, we can write: {S F(S)} 58

59 Continued F is called a formula (well-formed formula). For example, to express the query Find the staffno, fname, lname, position, DOB, salary, and branchno of all staff earning more than 10,000, we can write: {S Staff(S) S.salary > 10000} S.salary means the value of the salary attribute for the tuple variable S. To retrieve a particular attribute, such as salary, we would write: {S.salary Staff(S) S.salary > 10000} 59

60 The Existential and Universal Quantifiers There are two quantifiers we can use with formulae to tell how many instances the predicate applies to. The existential quantifier ( there exists ) is used in formulae that must be true for at least one instance, such as: Staff(S) ( B) (Branch(B) (B.branchNo = S.branchNo) B.city = London ) This means, There exists a Branch tuple that has the same branchno as the branchno of the current Staff tuple, S, and is located in London. The universal quantifier ( for all ) is used in statements about every instance, such as: ( B) (B.city Paris ) This means, For all Branch tuples, the address is not in Paris. 60

61 Continued Tuple variables that are qualified by or are called bound variables, otherwise the tuple variables are called free variables. The only free variables in a relational calculus expression should be those on the left side of the bar ( ). For example, in the following query: {S.fName, S.lName Staff(S) ( B) (Branch(B) (B.branchNo = S.branchNo) B.city = London )} S is the only free variable and S is then bound successively to each tuple of Staff. 61

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64 Domain Relational Calculus 64

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68 Chapter Summary 68

69 Recommended Readings Chapter 4 from: - Database Systems-A Practical Approach to Design, Implementation and Management by Thomas Connolly and Carolyn BEGG, 4 th Edition (Page No ) 69

70 Lecture 5 Summary of Lecture Relational Algebra 1. Unary Operations: 1) Selection Operation 2) Projection Operation 2. Set Operations: 1) Union Operation 2) Set Difference Operation 3) Intersection Operation 4) Cartesian Product Operation 70

71 Lecture 5 Summary of Lecture 3. Join Operations: 1) Theta/Equi Join Operation 2) Natural Join Operation 3) Left Outer Join Operation 4) Semi Join Operation 4. Division Operations: 1) Division Operation 5. Aggregate & Grouping Operations: 1) Aggregate Operation 2) Grouping Operation 71

72 Lecture 5 Summary of Lecture Relational Calculus 1. The Existential and Universal Quantifiers 2. Tuple Relational Calculus 3. Domain Relational Calculus 72

73 END OF LECTURE 5 73